Bayesian Networks Node Definition

Question

The educational content online for Bayesian Networks is not the best. (It's a subtle topic which leads to subtle questions and I'm having a hard time understanding it.)

It is my understanding that every node of a Bayesian Network is a probability distribution and that a node is conditionally independent of its non-descendants given its parents. Is this correct? Can nodes have multivariate distributions? If so, doesn't this mean that every statistical model can be represented as a one-node "bayesian network"? (a trivial one for sure, but still)

Answer 1

Your understanding is correct: in a Bayesian network, each node represents a random variable, which can have a multivariate probability distribution. The main reason to use a Bayesian network is that you have information (or other reasons) that helps you formulate conditional independence between certain variables, given others. You could definitely represent any number of random variables as a one-node Bayesian network, where the one node describes the full joint probability distribution over all the variables. In practice, you obviously wouldn't do this because it would generally be computationally intractable to work with a full joint distribution over many variables.

Hope this helps!

P.S. not really an answer to your question, but I personally found the Coursera specialization on Probabilistic Graphical Models to be quite informative. You can find it here: https://www.coursera.org/specializations/probabilistic-graphical-models